Let equation of hyperbola is a2x2−b2y2=1
∴ It passes through (4,−23)⇒a216−b212=1
⇒16−12×b2a2=a2...(1)
Equation of directrix is x=ea=54, given in question.
⇒a2=516e2...(2)
And we know that b2=a2(e2−1)
⇒a2b2=e2−1...(3)
∴ From (1),(2)&(3)
16−e2−112=516e2
⇒16e2−16−12=516e2(e2−1)
⇒4e4−24e2+35=0